Given an integer \(n\ge 2\) , the gauge-invariant NTM action for the field \(h_{\alpha (n)}\) 
given in [1]} is
\(S_{\text{NTM}}^{(n)}[h]=\frac{\text{i}^n}{2^{\lceil n/2 \rceil +1}}\frac{1}{\rho }\int \text{d}^3x\, e \, \mathfrak {C}^{\alpha (n)}(h) \big (\mathcal {F}-\rho \big )h_{\alpha (n)}~, \) 
